8.2 Multiply and Divide Rational Expressions

Rational expressions are like algebraic fractions on steroids. They involve polynomials in both the top and bottom, making them trickier to work with. But don't worry, we've got some handy tricks up our sleeves.

When multiplying or dividing these expressions, we follow similar steps to regular fractions. We multiply across the top and bottom, factor everything out, and then cancel common terms. It's all about simplifying to get the cleanest result possible.

Multiplying Rational Expressions

Multiplication of rational expressions

Top images from around the web for Multiplication of rational expressions

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Simplifying Rational Expressions View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Multiplying and Dividing Rational ... View original

  Is this image relevant?

• 

  Multiply and Divide Rational Expressions – Intermediate Algebra View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Simplifying Rational Expressions View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Multiplying and Dividing Rational ... View original

  Is this image relevant?

1 of 3

Top images from around the web for Multiplication of rational expressions

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Simplifying Rational Expressions View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Multiplying and Dividing Rational ... View original

  Is this image relevant?

• 

  Multiply and Divide Rational Expressions – Intermediate Algebra View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Simplifying Rational Expressions View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Multiplying and Dividing Rational ... View original

  Is this image relevant?

1 of 3

• Multiply numerators together combine like terms ($3x \cdot 2x = 6x^2$)
• Multiply denominators together combine like terms ($5y \cdot y = 5y^2$)
• Use product rule of exponents when multiplying variables with exponents ($x^2 \cdot x^3 = x^5$)
• Factor numerator and denominator completely after multiplying ($(x+1)(x-2)$ and $(x+3)(x-4)$)
• Cancel out common factors in numerator and denominator ($\frac{(x+1)\cancel{(x-2)}}{(x+3)\cancel{(x-2)}} = \frac{x+1}{x+3}$)
• Multiply any remaining factors to get final simplified expression ($\frac{2x}{3} \cdot \frac{3x}{4} = \frac{6x^2}{12} = \frac{x^2}{2}$)
• Ensure the result is in lowest terms through simplification

Dividing Rational Expressions

Division of rational expressions

• Multiply first expression by reciprocal of second expression ($\frac{2x}{3} \div \frac{4}{5x} = \frac{2x}{3} \cdot \frac{5x}{4}$)
• Reciprocal swaps numerator and denominator ($\frac{4}{5x}$ becomes $\frac{5x}{4}$)
• Multiply numerators together ($2x \cdot 5x = 10x^2$)
• Multiply denominators together ($3 \cdot 4 = 12$)
• Factor numerator and denominator completely ($10x^2$ and $12$)
• Cancel out common factors ($\frac{10\cancel{x^2}}{12} = \frac{5\cancel{x}}{6}$)
• Multiply remaining factors for final simplified expression ($\frac{5x}{6}$)

Complex fractions with rational expressions

• Contain fractions in numerator, denominator, or both ($\frac{\frac{2}{x}}{\frac{3}{x^2}+\frac{1}{x}}$)
• Multiply numerator and denominator by LCD of all fractions ($x^2$)
• LCD is smallest common multiple of all denominators ($x$ and $x^2$ gives $x^2$)
• Distribute LCD to each term in numerator ($\frac{2}{x} \cdot x^2 = 2x$)
• Distribute LCD to each term in denominator ($\frac{3}{x^2} \cdot x^2 = 3$ and $\frac{1}{x} \cdot x^2 = x$)
• Combine like terms and simplify numerator ($2x$)
• Combine like terms and simplify denominator ($3+x$)
• Resulting simplified complex fraction ($\frac{2x}{3+x}$)
• Further simplify by dividing numerator and denominator by common factors if possible

Working with Rational Expressions

• A rational expression is an algebraic fraction where both numerator and denominator are polynomials
• Always consider domain restrictions when simplifying rational expressions
• Simplification involves factoring and canceling common terms in numerator and denominator
• The goal is to express the rational expression in its simplest form or lowest terms